Annual Report 2008.pdf - SAMSI

“Robust Prediction Error Criterion for Pareto Modeling of Upper Tails”
Estimation of the Pareto tail index from extreme order statistics is an important problem in many
settings: income distributions, finance, and insurance. The upper tail of the distribution, where
data are sparse, is typically fitted with a model such as the Pareto model from which quantities
such as probabilities associated with extreme events are deduced. The success of this procedure
relies heavily not only on the choice of the estimator for the Pareto tail index but also on the
procedure used to determine the number k of extreme order statistics used for the estimation. We
develop and investigate a robust prediction error criterion to choose k and estimate the Pareto
index. The analysis of real data sets shows that a robust procedure for selection, and not just for
estimation, is needed.
Vicky Fasen
Munich University of Technology
fasen@ma.tum.de
“Extremes of Autoregressive Threshold Processes”
We study the tail and the extremal behavior of stationary solutions of autoregressive threshold
(TAR) models. It is shown that a regularly varying noise sequence leads only to an O-regularly
varying stationary solution in general. Under further conditions on the partition, it is however
shown that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided the
noise sequence is regularly varying. In these cases, the stationary solution is even multivariate
regularly varying and its extremal behavior is studied via point process convergence. In
particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in
contrast to TAR models with noise in the maximum domain of attraction of the Gumbel
distribution and which is either subexponential or in L(gamma) with gamma > 0. In that case it
turns out that the tail of the stationary solution behaves like a constant times that of the noise
sequence, regardless of the order and the specific partition of the TAR model, and that the
process cannot exhibit clusters on high levels.
Dominik Lambrigger
Eidgenossische Technische Hochschule Zurich
dominik.lambrigger@math.ethz.ch
“Multivariate Extremes and the Aggregation of Dependent Risks: Examples and Counter-
Examples”
Properties of risk measures for extreme risks have become an important topic of research. In the
present paper we discuss sub- and superadditivity of quantile based risk measures and show how
multivariate extreme value theory yields the ideal modeling environment. Numerous examples
and counter-examples highlight the applicability of the main results obtained.
Ross Leadbetter
University of North Carolina, Chapel Hill